Abstract
We consider second-order optimality conditions for set-valued optimization problems subject to mixed constraints. Such optimization models are useful in a wide range of practical applications. By using several kinds of derivatives, we obtain second-order necessary conditions for local Q-minimizers and local firm minimizers with attention to the envelope-like effect. Under the second-order Abadie constraint qualification, we get stronger necessary conditions. When the second-order Kurcyusz–Robinson–Zowe constraint qualification is imposed, our multiplier rules are of the Karush–Kuhn–Tucker type. Sufficient conditions for firm minimizers are established without any convexity assumptions. As an application, we extend and improve some recent existing results for nonsmooth mathematical programming.
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References
Abadie, J.: On the Kuhn–Tucker theorem in Nonlinear Programming (NATO Summer School, Menton, 1964). North-Holland, Amsterdam (1967)
Bannans, J.F., Cominetti, R., Shapiro, A.: Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9, 466–492 (1999)
Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990)
Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21, 151–176 (2013)
Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23, 632–665 (2013)
Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems. Math. Program. 41, 73–96 (1988)
Penot, J.P.: Second order conditions for optimization problems with constraints. SIAM J. Control Optim. 37, 303–318 (1999)
Gutiérrez, C., Jiménez, B., Novo, N.: On second order Fritz John type optimality conditions in nonsmooth multiobjective programming. Math. Program. Ser. B 123, 199–223 (2010)
Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49, 123–144 (2004)
Khanh, P.Q., Tuan, N.D.: Optimality conditions for nonsmooth multiobjective optimization using Hadamard directional derivatives. J. Optim. Theory Appl. 133, 341–357 (2007)
Khanh, P.Q., Tuan, N.D.: Second order optimality conditions with the envelope-like effect in nonsmooth multiobjective programming II: optimality conditions. J. Math. Anal. Appl. 403, 703–714 (2013)
Khanh, P.Q., Tuan, N.D.: Second-order optimality conditions with the envelope-like effect for nonsmooth vector optimization in infinite dimensions. Nonlinear Anal. 77, 130–148 (2013)
Taa, A.: Second order conditions for nonsmooth multiobjective optimization problems with inclusion constraints. J. Glob. Optim. 50, 271–291 (2011)
Zhu, S., Li, S.: Optimality conditions of strict minimality in optimization problems under inclusion constraints. Appl. Math. Comput. 219, 4816–4825 (2013)
Durea, M.: First and second order Lagrange claims for set-valued maps. J. Optim. Theory Appl. 133, 111–116 (2007)
Jahn, J., Khan, A.A., Zeilinger, P.: Second-order optimality conditions in set optimization. J. Optim. Theory Appl. 125, 331–347 (2005)
Li, S.J., Zhu, S.K., Li, X.B.: Second order optimality conditions for strict efficiency of constrained set-valued optimization. J. Optim. Theory Appl. 155, 534–557 (2012)
Khan, A.A., Tammer, C.: Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization 62, 743–758 (2013)
Zhu, S.K., Li, S.J., Teo, K.L.: Second-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization. J. Glob. Optim. 58, 673–679 (2014)
Khanh, P.Q., Tung, N.M.: First and second-order optimality conditions without differentiability in multivalued vector optimization. Positivity 19, 817–841 (2015)
Khanh, P.Q., Tung, N.M.: Second-order optimality conditions with the envelope-like effect for set-valued optimization. J. Optim. Theory Appl. 167, 68–90 (2015)
Studniarski, M.: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim. 24, 1044–1049 (1986)
Jiménez, B.: Strict efficiency in vector optimization. J. Math. Anal. Appl. 265, 264–284 (2002)
Flores-Bazán, F., Jiménez, B.: Strict efficiency in set-valued optimization. SIAM J. Control Optim. 48, 881–908 (2009)
Penot, J.P.: Differentiability of relations and differential stability of perturbed optimization problems. SIAM J. Control Optim. 22, 529–551 (1984)
Ward, D.E.: A chain rule for first and second order epiderivatives and hypoderivatives. J. Math. Anal. Appl. 348, 324–336 (2008)
Khan, A.A., Tammer, C., Zalinescu, C.: Set-Valued Optimization: An Introduction with Application. Springer, Berlin (2014)
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Applications, vol. II. Springer, Berlin (2006)
Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)
Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)
Khanh, P.Q., Kruger, A.K., Thao, N.H.: An induction theorem and nonlinear regularity models. SIAM J. Optim. 25, 2561–2588 (2015)
Ha, T.X.D.: Optimality conditions for several types of efficient solutions of set-valued optimization problems. In: Pardalos, P., Rassis, ThM, Khan, A.A. (eds.) Chapter 21, Nonlinear Analysis and Variational Problems, pp. 305–324. Springer, Heidelberg (2009)
Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, New Jersey/London (2002)
Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)
Ursescu, C.: Multifunctions with closed convex graph. Czechoslov. Math. J. 25, 438–441 (1975)
Jahn, J.: Introduction to the Theory of Nonlinear Optimization, 2nd edn. Springer, Berlin (1996)
Robinson, S.M.: Stability theorems for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)
Acknowledgments
This work was supported by the Vietnam National University Hochiminh City under Grant Number B2015-28-03. A part of the work was completed during a scientific stay of the authors at Vietnam Institute for Advance Study in Mathematics (VIASM), whose hospitality is appreciated. The authors are very grateful to the editors and referees for their valuable comments and suggestions.
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Communicated by Antonino Maugeri.
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Khanh, P.Q., Tung, N.M. Second-Order Conditions for Open-Cone Minimizers and Firm Minimizers in Set-Valued Optimization Subject to Mixed Constraints. J Optim Theory Appl 171, 45–69 (2016). https://doi.org/10.1007/s10957-016-0995-x
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DOI: https://doi.org/10.1007/s10957-016-0995-x
Keywords
- Set-valued optimization
- Abadie constraint qualification
- Kurcyusz–Robinson–Zowe constraint qualification
- Open-cone minimizer
- Firm minimizer